Proposition X.73 — If from a rational straight line there be subtracted a ratio…

Proof

a - b

with

a

b

commensurable in square only is the negation of the binomial case (X.36); the same argument shows it is irrational.

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Depends on (1)

X.36Proposition X.36If two rational straight lines commensurable in square only be added together, the whole is irrational; and let it be…

Required by (dependents) (7)

X.74Proposition X.74If from a medial straight line there be subtracted a medial straight line commensurable with the whole in square only,…
X.79Proposition X.79Only one rational straight line can be annexed to an apotome which is commensurable with the whole in square only.
X.85Proposition X.85To find the first apotome.
X.103Proposition X.103A straight line commensurable in length with an apotome is itself an apotome and the same in order.
X.108Proposition X.108If from a rational area a medial area be subtracted, the side of the remaining area arises as one of four irrationals:…
X.111Proposition X.111The apotome is not the same as the binomial.
XIII.6Proposition XIII.6If a rational straight line be cut in extreme and mean ratio, each of the segments is the irrational straight line…

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